Ngô Quốc Anh

March 5, 2014

Uniformly upper Bound for Positive Smooth Solutions To The Lichnerowicz Equation In R^N

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 16:16

In this note, we are interested in the following Lichnerowicz type equation in \mathbb R^n

\displaystyle -\Delta u =-u^q+\frac{1}{u^{q+2}}, \quad q>0.

As we have already seen from the previous note that solutions for the above equation are always bounded from below for certain q.

Theorem (Brezis). Any solution of the Lichnerowicz equation with q>0 satisfies u\geqslant 1 in \mathbb R^n.

Remarkably, we are able to prove that solutions for the Lichnerowicz type equation are also bounded from above if we require q>1 instead of q>0. This result is basically due to L. Ma and X. Xu, see this paper.

Theorem (Ma-Xu). Any solution of the Lichnerowicz type equation with q>1 is uniformly bounded from above in \mathbb R^n.

Combining the two theorem above, we conclude that any solution of the Lichnerowicz equation, i.e. q=(n+2)/(n-2), is uniformly bounded in \mathbb R^n.

The idea of the proof for Ma-Xu’s theorem is as follows: Denote

\displaystyle f(u)=u^{-q-2} - u^q.

Fix x_0 \in \mathbb R^n but arbitrary, we then look for a positive radial super-solution v(x)=v(|x|)>0 of the Lichnerowicz type equation on the ball B_R(x_0) with positive infinity boundary condition for some R to be specified. This is equivalent to finding v in such a way that

\displaystyle\begin{array}{rcl}\displaystyle-\Delta v &\geqslant& f(v), \qquad\text{ in } B_R(x_0),\\ v&\equiv& +\infty, \qquad\text{ on }\partial B_R(x_0).\end{array}

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